AL-QALA??D? (OR AL-QALA??D?), ABU ’L-?ASAN ‘AL? IBN MU?AMMAD IBN ‘AL?

(b. Basta [now Baza], Spain, 1412; d. Béja, Tunisia, December 1486)

arithmetic, algebra, Islamic law.

Al-Qala??di, the last known Spanish-Muslim mathematician, is also known by the epithets al-Qurash? and al-Bas??, the latter referring to his place of birth. He remained in Bast?? until the city was taken by the Christians, at which time he began his journey through the Islamic world and studied with learned men.

Several books on arithmetic and one on algebra are attributed to al-Qalas??di. The one on algebra is a commentary on the al-urj?z? al-Y?sm?n?yaof Ibn al-Y?sm?n? (d. 1204), which gave algebraic rules in verse. This urj?z? (poem) had been commented upon by several Western Muslims before al-Qala??di.

One of his arithmetical works is a commentary on the Talkhi? a’m?l al-?is?b (“Summary of Arithmetical Operations”) of Ibn at-Bann? several summaries and extracts of this commentary have reached us. Al-Qala??d?’ original work began with al-Tahsira fi’lm at-?is?b(“Clarification of the Science of Arithmetic”); it proved to contain difficult material, and he therefore simplified it in Kashf al-jilb?b an ilm al-Itisdb (“Unveiling the Science of Arithmetic”). A shorter version of the same work is Kashf al-asr?r an ’ilm al?is?b (“Unfolding the Secrets of the Use of Dust Letters [Hindu Numerals]”)The last two are said to have been used in some schools of North Africa for several generations, and the last is the work studied by F. Woepcke in his “Traduction du traite d’ ariithmetique d’Aboul-Hacan Al? Ben Mohammed Alklcadi” (Atti dell’ Accademia Pontificia d’ nuovi lincei),12 [1858–1859], 399–438)

Since the 1850s’ al-Qala??d? has been credited with the following:

Dealing with the sequences ?n² and ?n³.

Using the method of successive approximations for obtaining the roots of imperfect squares.

Using symbols in algebraic equations.

In the light of our present knowledge, the following can be stated:

First, al-Qala??di has claim to priority in the field of sequences. Those that he treated and more advanced ones, such as those of polygonal and pyramidal numbers, had been treated by Ab? Mansur al-Baghada?d? (d1037) and al-Umaw? al-Andalus? (ft. fourteenth century).

Second, the method of finding square roots by successive approximation had been known to the Greeks and, probably, the Babylonians. In principle, it states that if r₁ is an approximation ?n,of then let r₂ = n/r₁ and a better approximation is r₃ = l/2(r₁ + r₂).

This method must have been known to the arithmetician of eastern Islam, but they seem to have preferred to find roots, expressed in sexagesim?l fractions in almost the same way we use to find them to any desired decim?l place. Al-Qala??di, however, is the first mathematician known to have stressed it.

Third, al-Qala??di used both short Arabic words and letters as symbols. The short words are

wa(and) for addition

ill?(less) for subtraction

fi(times) for multiplication

’al?(over) for division.

Letters were also used to designate certain terms; these correspond to:

jfor jadhr(root)

sh (from shay, “thing”), or (the diacritical points from shay) for x.

mfor m?l(x²)

k for ka?b (x³)

mm for mal mal(x⁴)

l(from the verb ya?dilu) for equali?ty.

The letters. more than the words, indicate a sense of symbolism. But al-Qala??di has no claim to priority here, either; the same symbols were used in the same way by Ibn Qunfudh of Algiers(d 1407/1408) and Ya’qyb Ibn Ayyub of Morocco (fl. ca. 1350), and many earlier writers in the East.

Like similar works from the thirteenth century on, al-Qala??di’ writings show Arabic arithmetic and algebra when their constituents-ancient manipulational tradition, Hindu techniques, and Greek number theory—-are combined to form one entity. But they also reflect a civilization on the wane, for most of them are commentaries, summaries, or summaries of summaries of works by al-Qala??di himself or by others.

BIBLIOGRAPHY

See C. Brockelmann, Geschichte der arabischen literatur, II, pt. 2 (Leiden, 1949), 343–344, and supp. II (Leiden, 1938), 363–369; al-Maqqari, Nafh al-tib, Ihsan ’Abbas, ed., II (Beirut, 1968), 692–694; and H. Suer, Die Mathematiker und Astronomen der Araber und ihre Werke(Leipiz, 1900) no. 444, pp. 180–182. The best Arabic bilography of al-Qala??di is perhaps that written by M. Souissi of the University of Tunisia in the University Periodical, 9 (1972), 33–49.

A. S. Saidan